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<p>In this paper, we introduce two new classes of bounded linear operators viz. the classes of strongly minimum norm attaining operators and absolutely strongly minimum norm attaining operators, both contained in the class of minimum norm attaining operators. We prove that if <i>Y</i> is an LUR space, then for any Banach space <i>X</i>, every quasi-minimum norm attaining operator from <i>X</i> into <i>Y</i> can be approximated by a strongly minimum norm attaining operator from <i>X</i> into <i>Y</i>. We also show that the class of all absolutely strongly minimum norm attaining operators between any two Banach spaces <i>X</i>,&#xa0;<i>Y</i> is a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G_{\delta }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>δ</mi> </msub> </math></EquationSource> </InlineEquation> subset of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {B}(X,Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We define a weaker form of the approximate minimizing property for a pair of Banach spaces, called the weak approximate minimizing property and define a class <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {WAM}(X,Y)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">WAM</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with the help of this property.</p>

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Attainment of the minimum norm and Bishop-Phelps-Bollobás type properties

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Rajarshi Saraswati,
  • Uday Shankar Chakraborty

摘要

In this paper, we introduce two new classes of bounded linear operators viz. the classes of strongly minimum norm attaining operators and absolutely strongly minimum norm attaining operators, both contained in the class of minimum norm attaining operators. We prove that if Y is an LUR space, then for any Banach space X, every quasi-minimum norm attaining operator from X into Y can be approximated by a strongly minimum norm attaining operator from X into Y. We also show that the class of all absolutely strongly minimum norm attaining operators between any two Banach spaces X, Y is a \(G_{\delta }\) G δ subset of \(\mathcal {B}(X,Y)\) B ( X , Y ) . We define a weaker form of the approximate minimizing property for a pair of Banach spaces, called the weak approximate minimizing property and define a class \(\mathcal {WAM}(X,Y)\) WAM ( X , Y ) with the help of this property.